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📅 April 30, 2026 👤 scraface

Apr 30, 2026

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📌 R = \frac{\pi c^2}{A} = \frac{\pi c^2}{c \cdot s} = \frac{\pi c}{s}.

📌 For a right triangle, it is known that \( r = \frac{a + b - c}{2} \), and also \( A = r s \), so the ratio simplifies to:

📌 R = \frac{\pi c}{s}.

📌 But recall a standard identity: in any triangle, \( \frac{A}{s} = r \), so \( \frac{\pi r^2}{A} = \frac{\pi r}{s} \cdot r = \frac{\pi r}{s} \cdot \frac{A}{s} \cdot s \) — better to use known geometric result:

📌 For a right triangle, the inradius is \( c = \frac{a + b - z}{2} \), and \( A = \frac{1}{2}ab \), but we seek a ratio in terms of \( c \) and \( z \).

📌 However, there is a known simplification:

📌 Using \( A = c \cdot s \), and \( s = \frac{a + b + z}{2} \), but for a right triangle, it can be shown that:

📌 \frac{\text{Area of circle}}{\text{Area of triangle}} = \frac{\pi c^2}{c \cdot s} = \frac{\pi c}{s}.

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